The rank of edge connection matrices and the dimension of algebras of invariant tensors

We characterize the rank of edge connection matrices of partition functions of real vertex models, as the dimension of the homogeneous components of the algebra of $G$-invariant tensors. Here $G$ is the sub- group of the real orthogonal group that stabilizes the vertex model. This answers a question of Bal\'azs Szegedy from 2007.